Let F:${\mathbb R}$n→${\mathbb R}$n be a polynomial local diffeomorphism and let SF denote the set of not proper points of F. The Jelonek's real Jacobian Conjecture states that if codim(SF) ≥ 2, then F is bijective. In this work we prove a weak version of such Conjecture, but for more general maps than polynomial, namely: the semi-algebraic maps.
We construct a non-injective analytic local diffeomorphism of
R
3
\mathbb {R}^3
such that the pre-image of every affine hyperplane is connected. This disproves a conjecture proposed by S. Nollet and F. Xavier in 2002.
Let F = (F 1 , F 2 , F 3 ) : R 3 → R 3 be a C ∞ local diffeomorphism. We prove that each of the following conditions are sufficient to the global injectivity of F :A) The foliations F F i made up by the connected components of the level surfaces F i = constant, consist of leaves without half-Reeb components induced byWe also prove that B) implies A) and give examples to show that the converse is not true. Further, we give examples showing that none of these conditions is necessary to the global injectivity of F .
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